I lived there for a year, only towards the end of it did I find anyone that rides in Leamington (other than at the 4X track that is).
Best bet is to get down to John Atkins Cycles, and have a chat with the guys in there. I’m sure they’ll be able to point you in the direction of some guys that go riding regularly.
Alternatively, if he can be bothered to get up to Coventry, the guys from Godiva Trail Riders are a great bunch. Again, I only found them a couple of months before I left Cov, but for the 2 months I rode with them they showed me lots of trails. They ride every Wednesday evening pretty much, and usually stop and have a beer in the pub afterwards.
http://www.godivatrailriders.org/